Problem 826

Birds on a Wire

Consider a wire of length $1$ unit between two posts. Every morning $n$ birds land on it randomly with every point on the wire equally likely to host a bird. The interval from each bird to its closest neighbour is then painted.

Define $F (n)$ to be the expected length of the wire that is painted. You are given $F (3) = 0.5$ .

Find the average of $F (n)$ where $n$ ranges through all odd prime less than a million. Give your answer rounded to $10$ places after the decimal point.

电线上的小鸟

两根电线杆之间有长为 $1$ 的电线。每天早上， $n$ 只小鸟均匀随机地落在这段电线上。将每只小鸟和其最近邻居之间的电线染上颜色。

记 $F (n)$ 为被染色电线的总长度的期望值。已知 $F (3) = 0.5$ 。

当 $n$ 取遍所有小于一百万的奇素数时，求所有 $F (n)$ 的平均值。将你的答案四舍五入至小数点后 $10$ 位。